WORKING PAPER N° 2007 - 08

Technical progress in North and welfare gains in South

under nonhomothetic preferences

Lilas Demmou

JEL Codes: F11, O11 Keywords: Dornbush-Fisher-Samuelson Ricardian

model, technology and trade, North-South trade, nonhomothetic preferences, hierarchic needs, hierarchic purchases

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Technical progress in North and welfare gains in Southunder nonhomothetic preferences

Demmou Lilas1

February 2007

1 l.demmou@yahoo.fr

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Abstract

The paper proposes a theoretical model investigating the welfare consequences of tech-nological shocks in a Ricardian framework (a la Dornbush, Fisher and Samuelson, 1977).Contrary to existing literature, the model incorporates a nonhomothetic demand func-tion whose price and income elasticities are endogenously determined by technology. Themodel is applied to the case of trade between two economies with di¤erent developmentlevels. It is shown in particular that the developing country can experience a fall in utilityas a result of technical progress in the developed country. This result depends on the typeof technological shock assumed (biased vs. uniform technical progress), as well as on thesize of the development gap.

Keywords : Dornbush-Fisher-Samuelson Ricardian model ; Technology and Trade ;North-South Trade ; Nonhomothetic preferences ; Hierarchic needs ; Hierarchic purchases

JEL Classi…cation : F11, O11

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1 IntroductionThe role of nonhomothetic preferences in the analysis of trade has experienced a sharp risein importance in the recent literature. This assumption about consumption behavior hasbeen included in a bunch of papers dealing with di¤erent questions such as the predictedfactor content of trade (Dinopoulos, Fujiwara and Shimomura (2006), Chung (2003)), thee¤ects of trade liberalization or preferential trade agreements on welfare (Stibora andde Vaal (2007), Stibora and de Vaal (2006)), the relation between income inequality andtrade ‡ows (Mitra and Trinidade (2003), Kugler and Zweimüller (2002)) and the in‡uenceof technological progress on the distribution of gains from trade (Matsuyama (2000)).

The reason for including nonhomothetic preferences in a trade model, despite the lowertractability of this kind of function, is that there is much evidence to show that deviationfrom homothetic preferences is not marginal. One illustration of this is provided by amajor …nding in the empirical literature devoted to this issue: Hunter (1991) proposesa counter-factual analysis and shows that nonhomothetic preferences account for 25% ofworld trade. In the same line, several empirical contributions prove that income elastic-ities in trade are actually non unitary (see, for instance, Magee and Houthakker (1969),Atesoglu (1993, 1994); Bairam, Dempster (1991); Perraton (2003) and Muscatelli et al.(1994, 1995)). However, most contributions in the existing literature are based on theassumption of homothetic preferences. This raises the question of whether certain im-portant theoretical results in trade literature are robust to changes in the speci…cation ofpreferences.

This paper is devoted to the old question of the international distribution of welfaregains induced by technical progress. More speci…cally, I look at the relative ability of adeveloping country to bene…t (in terms of welfare) from the external technical progressthrough trade. Among the large literature devoted to this issue, the standard approach isthe Ricardian framework : according to Dornbush, Fisher and Samuelson (hereafter, DFS(1977)), technical progress in one country always bene…ts the trading partner, through aterms-of-trade deterioration for the fast-growing country. This theoretical result relies onthe use of a Cobb Douglas utility function yielding unitary income and price elasticities intrade. It prevails in major inter-sectorial trade models such those of endogenous growth(see for instance, Grossman and Helpman, chapter 7, 1991).

I propose here a ricardian model where non unitary elasticities in trade are microfundedon the basis of non homothetic preferences1 . This model is related to the Ricardian inter-sectorial trade models of Matsuyama (2000)2 . Actually, Matsuyama (2000) proposes anonhomothetic model which is considered as an alternative to the standard homotheticmodel of DFS (1977). Matsuyama models nonhomothetic preferences through a functionof hierarchic desires, where agents extend the range of consumed goods when real incomeincreases. As the quantities of each good consumed are exogenously …xed to one, theevolution of consumption can only be extensive. The common point shared by the Mat-suyama and DFS models is that they …x, a priori, the evolution of consumption patternsin the event of technical progress. In the DFS model, demand shifts towards those goodswhich display the greatest decrease in relative price (so as to maintain the initial distribu-tion of spending on each good). In Matsuyama’s approach, the change in demand favoursthe lower priority goods produced by North.

By allowing agents to choose both the quantities and the range of goods consumed, I1 The theoretical interpretation of non-unitary income elasticities in trade is not straightforward. Ac-

cording to McGregor and Swales (1986, 1991) and Krugman (1989), this result is a statistical artefactresulting from a supply composition e¤ect. On the other hand, nonhomothetic preferences occupy acentral position in the post-Keynesian analysis of income elasticities of trade (Thirlwall (1979), Kaldor(1970)). This debate lies outside the purposes of this paper. The choice of a nonhomothetic preference-based approach is found in some empirical studies reported in section 2.1. More details can also be foundin studies which control for the supply side composition e¤ect in their estimations of export and im-port functions: Amable (1992), Fagerberg (1988), Feenstra (1994), Gagnon (2003), Funke and Runwedel(2001), Bayoumi (1999).

2 We can also refer to the nonhomothetic intra-sectorial models of Flam-Helpman (1987) and Stockey(1991), which are discussed in detail in Matsuyama (2000).

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propose a new framework to account for non homothetic preferences alternative to Mat-suyama (2000). The main speci…city of my approach is the link established between theprice and income elasticities of external demand, and the nature of the goods in whicheach country is specialized. This link forwards by two transmission channels. Firstly,I consider hierarchic consumption behavior. Goods are then distinguished according totheir relative degree of priority, which also determines their respective income and priceelasticity values. Secondly, the technological dimension is linked to the consumption be-havior through a function of hierarchic purchases. In particular, I allow technical progressto change the relative degree of priority of each good. The intuition behind this is thattechnical progress, by the creation of new goods, leads to the creation of new needs, andthen modi…es the perception of agents regarding their pre-existing needs. This e¤ect isformally expressed by the fact that elasticities are partially determined by technical coef-…cients of production. Hence, technical progress has a direct impact on the distributionof spending devoted to each good and on the range and quantities of goods consumed.

The aim of the paper is then to analyze the impact of technical progress on the interna-tional distribution of welfare, in a framework where hierarchic consumption, non-uniformsubstitution and income e¤ects are simultaneously included. Our main result is that Southcan be harmed by technical progress in the North. This happens if technical progress isbiased toward lower priority goods and if the size of the technological gap between thetwo countries is large. The decrease in South’s welfare occurs despite the fact that South’sagent consumes a wider range of goods produced by North. This result derives directlyfrom the link established between preferences and technology. It is totally at odds withresults from main inter-sectorial Ricardian models, where an increase in the existing pat-tern of comparative advantages leads to increased gains from trade (DFS (1977), Krugman(1990), Matsuyama (2000)). On the contrary, our result suggests that, for less advancedcountries to avoid being harmed by North’s technical progress, there exists an optimallevel of development gap with their trading partners. Due to the complexity of the model,I analytically study the case of a very large development gap between North and South.I also provide simulations for the more general case.

The rest of the paper is organized as follows. In the next section, a closed-economymodel and the main properties of the demand functions are presented. In the third section,these functions are incorporated into a Ricardian model of the type developed by DFS.A model of North-South trade is then obtained where the price and income elasticities ofexternal demand are non-unitary and micro-founded. The …nal section is devoted to theanalysis of the impact of North’s technical progress on South’s welfare.

2 A nonhomothetic closed-economy model2.1 Characteristics of consumption behaviorA series of macro- and microeconomic empirical studies on demand behavior providesdirect validation for the hypothesis of nonhomothetic preferences. We can notably con-sider the following evidences. Firstly, Falkinger, Zweimüller (1996) and Jackson (1984)show that a rise in income is accompanied by an extension in the range of goods con-sumed by agents. Secondly, Hunter (1991), Hunter-Markusen (1988), Jackson (1984) andFillat-Francois (2004), test the empirical relevance of some speci…c nonhomothetic de-mand functions (respectively, Stone Geary, Almost Ideal Demand System and hierarchicconsumption). They verify that when their income rises, agents do not distribute thisincrease uniformly over all the goods they buy3 . Finally, a corollary from the empiricalvalidity of the previous nonhomothetic demand functions is that sector income elasticitiesdepend on the agent’s level of income.

3 It should be noted that, as the Hunter (1991), Hunter-Markusen (1988) and Francois-Fillat (2004)studies rely on cross section data, they indirectly control for the impact of supply change composition ondemand income elasticities. One can also refer to the paper by Bills and Klenow (2000), who calculateEngel’s curve by controlling for quality.

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We require a demand function which is consistent with these three previous stylizedfacts: hierarchic consumption, non unitary income elasticities at the sectorial level, and alink between demand income elasticities and agents’ levels of income. From a theoreticalpoint of view, our approach to preferences is then related to that of Jackson (1984) andHunter-Markusen (1988). They use a utility function of the following type:

U =X

βi log(qi + γi)di (1)

where qi corresponds to the quantity of good i consumed.This quasi-homothetic utility function can produce a linear expenditure system (LES)

or a hierarchic linear expenditure system (HLES), depending on the sign of the constantterms γ i. For instance, Hunter and Markusen (1988) consider negative constant terms.In this way, they account for the need to satisfy a minimum consumption of an exogenousnumber of sectors4 . Subsequently, the function produces (non unitary) income and sub-stitution e¤ects. On the contrary, Jackson (1984) assumes a positive sign for the constantterms: the utility function relies more on an endowment e¤ect, which produces hierarchicconsumption of an endogenous range of consumed goods. For some goods, the non neg-ativity constraint may e¤ectively become binding. The main interest of this approach isthat it allows both for extensive and intensive demand behavior5 .

We follow Jackson’s approach. In this way, we can simultaneously account for non uni-tary elasticities of demand at the sectorial level and hierarchic consumption. Nevertheless,we modify the previous utility function in three ways. Firstly, it is expressed in continuumso that we can obtain an expression for the marginal good. Secondly, the standard CobbDouglas coe¢cient βi is assumed to be equal to one. All goods in the utility function arethen symmetric, except as far as the endowment e¤ect is concerned. Thirdly, we make alinear transformation by dividing the term in brackets in equation 1 by γ i. The purposeof this last transformation is to obtain a function where only consumed goods enter intothe agent’s …nal utility6 . The individual utility function at the heart of our model canthen be written:

U =

1Z

0

log(qi

γ i+ 1)di (2)

The maximization utility program of the representative consumer can be written:

Max U s.t.

1Z

0

piqidi = y

and qi ¸ 0

where pi and y correspond respectively to the price of good i and to the agent’s income.U is given by equation 2.

We can write the Khun-Tucker conditions:

for i 2 K,1

piγi> λ => qi =

1piλ

¡ γ i (3)

for i /2 K,1

piγi< λ => qi = 0 (4)

for i = J,1

pJ γJ= λ => qi = 0 (5)

4 One could also refer to the approaches of Markusen (1986), Matsuyama (1992) and Puga-Venables(1999), who apply the minimum consumption framework in a model with two sectors.

5 Foellmi and Zweimüller (2006) propose a closed-economy model where the utility function presentsthe same characteristics.

6 Otherwise, according to equation 1, for all goods consumed, utility is determined by βi log(qi + γi),but non-consumed goods also enter into the agent’s utility through βi log(γi).Our transformation removesthis e¤ect.

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According to these conditions, a good will or will not be consumed depending on itsvalue for the term piγi . This term determines hierarchic consumption and can then beconsidered as an entry criterion. Ceteris paribus, the lower the value of this term, themore likely it is that the good will be consumed, and the higher the priority attachedto it. Agents will consume lower priority goods only if what are perceived as more basicneeds have been already satis…ed in terms of quantity (following the equation 3)7.

To order the consumption process explicitly and determine the marginal good J , wehave to assume that the entry criterion follows an increasing monotonic function. Thisassumption enables us to divide the continuum into two segments: that of consumedgoods i 2 K = [0, J [ and that of non-consumed goods i /2 K = [J, 1[ (…gure 1).

We can then write the Lagrange multiplier:

λ =

Z

i2Kdi

y +JR0

piγidi(6)

Using equations 5 and 6, we can determine an implicit equation in J for the extensivedemand function. Similarly, on the basis of equations 3 and 6 and simplifying by equation7, we obtain an intensive demand function (qi):

J =y +

JR0

piγ idi

pJ γJ(7)

piqi = pJγJ ¡ piγi if piγi < pJγJ (8)piqi = 0 if piγi > pJ γJ

In addition, using equations 3 and 6 and deriving the induced demand function, weobtain an expression for demand income elasticity. In the same way, we can also deducetwo useful expressions for own-price and cross-price elasticities.

ηiy =

1J

.y

piqi(9)

ηip = ¡1 ¡ γi

qi.µ

1 ¡ 1J

¶< ¡1

ηip = ¡1 ¡

µpJ γJ

piqi¡ 1

¶.µ

1 ¡ 1J

¶(10)

ηipk

=1J

.pkγk

piqi> 0 (11)

We now present changes in consumption behavior according to changes in income (y)and in the ranking of the good in the continuum (i). The main properties of our demandfunctions are summarized by seven theorems reported in table 1. We brie‡y present theproofs for these theorems in annex 1.

From these theorems, we can highlight that in our model, the heterogeneity of agents’consumption baskets depends partly on their levels of income, and partly on the charac-teristics of the goods supplied (if it has higher or lower priority). Changes in consumption

7 On this point, we should explain why we have chosen to maintain sectorial di¤erences for the γ term.By applying the same value for γi , the entry criterion would only depend on relative prices (i.e. on unitvalues). We wanted to avoid this e¤ect, as the model refers to an inter-sectorial approach. E¤ectively,in this case, units are likely to vary from one sector to another, and there is no justi…cation for choosingunit values as an entry criterion. The weighting with γi allows us to get around this problem. The termγi then refers to a pure preference e¤ect, which determines the willingness of the agent to consume agood. In addition, multiplication by pi accounts for the possibility of consuming it according to the pricestructure (given the level of income). Because of this in‡uence of price on the composition of consumption,preferences re‡ect hierachic purchases rather than hierachic desires.

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behavior resulting from an increase in income take the form of access to what had initiallybeen non-priority (T1), together with a non uniform variation in the spending on eachgood (T2, T3). Based on equation 7, …gure 3 represents graphically the evolution of inten-sive and extensive consumption with income8 . Our extensive demand function presentsalso speci…c characteristics: income elasticities are in accordance with the stylized facts weaim to account for, i.e. non uniform evolution of expenditures between sectors and agents.More speci…cally, the last good to enter the consumption basket has the highest incomeelasticity of demand (T5). Thus, every good, except the …rst one always consumed, canbe considered as a "luxury good" the …rst time it enters the agent’s consumption bas-ket. With the entry of new goods, the previously-consumed goods tend to behave moreand more like necessity goods, as and when their weight in the consumption increases.Furthermore, the agents’ perceptions of the degree of priority of each good vary withtheir respective income levels. This is formally re‡ected by the fact that the lower theagent’s relative level of income, the more sensitive demand for good i will be to variationin income (T4).

Our extensive demand function presents two further main characteristics concerningprice elasticities. As with income elasticities, a good’s level of sensitivity to price variationsdepends on its novelty. Agents tend to adapt their consumption in favour of the lowerpriority goods when there is an uniform fall in prices, and to the detriment of these samegoods when there is a rise in prices (T6). These increasing values for price elasticities alongthe continuum seem reasonable: the higher the good’s priority, the less it is substitutable.Finally, according to the cross-price elasticity expression, the substitution e¤ect is higherthan the income e¤ect. In addition, the lower the good’s priority, the more its demand issensitive to the price variation of another good (T7).

The ranking of goods in the continuum according to their degree of priority and the pre-vious properties of our model both hold under the assumption of an increasing monotonicfunction for the entry criterion piγ i. The supply side of the model is de…ned such asto verify this relation. In this way, we establish a link between preferences and technol-ogy : agents’ perceptions of the priority of each good appear to be dependant on thetechnological dimension.

2.2 The link between technology and preferencesThe supply side of the closed economy is modeled in a very simple way. We assume asingle-input (labor) production function with constant returns. There are no pro…ts inthe economy. Each agent’s income is then given by the wage level. The national incomecorresponds to total wages uniformly distributed over the population. We denote ai, thequantity of labor required to produce one unit of good i. We assume perfect competition,so that the prices of goods are given by wage (w) and the inverse of labor productivity(ai):

pi = aiw (12)

Without losing generality, we can assume that goods are ranked in the continuum inincreasing order, so that the parameters of the model satisfy the following relation:

aiγi increasing function of i (13)

Thus, using equation 12, piγ i is also an increasing function of i. All the properties ofthe consumption function described above are therefore maintained.

We should highlight the fact that the way technical progress will in‡uence equilibriumconsumption derives straightforwardly from our way of modeling the hierarchic consump-tion behavior. By applying equation 12 and proposition 13 to our HLES, we e¤ectively

8 It should be noted that the choice of a convex curve for piγi is arbitrary. Theorems are valid for thegeneral case.

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establish a link between technical progress and preferences. Technical progress, by modi-fying the values of aiγ i and the entry criterion piγ i, will impact on the range of consumedgoods, the share of spending devoted to each good, and their respective income and priceelasticities of demand (equations 7-11). Actually, our goods ranking, which combines atechnological criterion9 (ai) and a criterion based on the utility function (γ i) , expressesthe fact that a technological shock (via ai) or a preference shock (via γ i) have the sameimpact on the composition and evolution of consumption. This equivalence is supportedby two arguments. Firstly, technical progress and increases in real income are two sidesof the same phenomenon. Accordingly, every increase in income (or technical advance)involves changes in the distribution of expenditures between goods, because of Engel’slaw [Pasinetti, 1983, p. 69]. Secondly, the equivalence between a technological shock anda preference shock is based on the following intuition. When a technical advance resultsin the creation of new goods, it also entails the creation of new needs and modi…es agents’perceptions of pre-existing goods (in terms of needs)10 . We could say that the boundaryline between "essential" needs and "psychological" needs is modi…ed1 1 .

The mechanism of an endogenous change in consumption behavior with technicalprogress is at the heart of our theoretical results. Technical progress, by changing realincome, is likely to modify the range of consumed goods (theorem 1). But at the sametime, in our model, technical progress also induces a change in the threshold level of thequantities consumed necessary to extend the range of consumption (equation 8). Variationin utility therefore depends on the evolution of these two components, quantities and rangeof goods consumed, which may not change in the same way. This kind of endogenouschange in consumption is graphically represented in …gure 4.

At this point, it may be useful to compare our approach to preferences with that of DFSand Matsuyama. The di¤erences in demand behavior in the three models are presentedin table 2. Our model is clearly distinct from DFS, where there is an homothetic demandfunction of the Cobb Douglas type. This exogenously constraints the range of consumedgoods and the distribution of spending between goods. Nevertheless, the two models canconsidered close insofar as they both allow for endogenous determination of quantitiesconsumed. Our approach also appears to be closely related to that of Matsuyama (2000),based on an utility function of hierarchic desires1 2. In this case, agents consume one unitof each good according to an order of priority that is …xed a priori. As in our model, therange of consumed goods, and subsequently the share of spending devoted to each good,are endogenous. Nevertheless, his approach di¤ers from our own in that he only allowsfor an extensive change in consumption behavior: income and price elasticities are nullfor each good.

We now embed our demand function in a Ricardian trade framework. The di¤erencesdescribed above concerning the utility function of the three approaches will be re‡ectedby di¤erences in the mechanisms involved in the trade balance equilibrium.

9 Fixed capital is not explicitly introduced into the model. However, we consider that the input of laborrepresents e¤ective labor, i.e. a composite (comprising labor, capital, and human capital).

10 Like Young (1991), we have not modeled the innovation of goods in our model: an in…nite numberof goods are theoretically available for consumption, but some of them are too expensive to be produced.Technical progress makes it possible to produce these goods by reducing production costs.

11 We should stress the fact that in our model, no goods, except the …rst one, can be considered essentialin physiological terms. It is only the perception of the agents (according to income and price structure)which makes each of them relatively essential.

12 The function of hierarchic desires is often used to link consumption expenditures with income distri-bution (Murphy, Shleifer and Vishny (1989) or Zweimüller (2000)). This is also the case in Matsuyama(2000). To be able to compare his approach with our own, we only consider the characteristics of hismodel under the assumption of a representative agent (i.e. with an uniform income distribution).

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3 The nonhomothetic Ricardian model3.1 International specializations and consumptionWe consider two economies with di¤erent levels of development. The more developedeconomy is the foreign one, and it is denoted by an asterisk. The foreign wage is thenumeraire (w¤ = 1) and the domestic wage (w) then re‡ects the relative wage. The generalstructure of the open-economy model is established following the standard hypotheses ofDFS (1977). Firstly, it is assumed that competition in the two countries is perfect. Thereis therefore an international price pm

i , which corresponds to the trade equilibrium atpm

i = Min fpi, p¤i g . Secondly, the foreign economy is more developed in the sense that it

is more productive, in absolute terms, in every sector a¤i < ai. Thirdly, the productivity

advantage of the developed country increases along its goods continuum, i.e. aia¤

iis an

increasing function of i.We can demonstrate that under the hypotheses of a technological gap increasing along

the whole goods continuum of the developed country, and of an identical utility functionin the two countries, the ranking of goods (in terms of priority level) in the developedcountry entails an identical ranking in the developing country:

if∂a¤

i γ i

∂i> 0 and

∂ aia¤

i

∂i> 0 then

∂aiγi

∂i> 0

In other words, the ranking of goods established for the developed economy is an"international" ranking. We can therefore identify the comparative advantage of eachcountry in terms of the nature of goods respectively traded. Like Matsuyama (2000),we thus establish a relation between the characteristics of a country (its level of devel-opment) and the characteristics of the goods in which it specializes (relative degree ofpriority). Formally, the specialization equation which determines the segment of South’sspecializations

£0, i

¤and, which holds also in the models of Matsuyama and DFS, can be

written:

w =a¤

iai

(14)

Given the hypothesis of the technology gap, trade can only take place if, for all i 2£0, i

¤, p¤

i > pi . For the developing country to be competitive in this goods segment, theminimum condition is that w < 1. The developing economy will then be specialized inthe beginning of the goods continuum, i.e. in goods with higher priorities. These goodspresent a relatively lower technological gap and lower income elasticity of demand (seetheorem 5)13 .

Concerning consumption, the marginal goods consumed in the two countries can bedetermined using equation 7:

J =w +

JR0

pmi γ idi

γJpmJ

(15)

J ¤ =1 +

J¤R0

pmi γ idi

γJ¤pmJ¤

(16)

13 It can be noted that this characteristic allows us to interpret the degree of priority of each good interms of the degree of sophistication. Empirical studies provide evidence in favor of a positive relationbetween income level and the share of expenditures devoted to the importated technological, di¤erentiated,and higher quality goods (see respectively, Meliciani (2002), Francois and Kaplan (1996), Hallak (2003)).

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Because agents in the two countries face identical prices for goods, the di¤erence inrelative wage expresses a di¤erence in real income. We can therefore apply the maintheorems de…ned in the closed economy.

Corol lary 1 : The developed country consumes more products and these products arelower priority.

E¤ectively, using theorem 1 and relation 13, and given that the relative wage is lessthan one, we can write J ¤ > J and therefore pm

J¤ γJ¤ > pmJ γJ .

According to this corollary and the previous assumptions about the technological di-mension, the structure of consumption and specializations induced by the model can besummarized as in table 314 .

3.2 Trade equilibriumWe make the standard assumption of a balance of trade equilibrium constraint whereX and M represent respectively the exports of the developing country (imports of thedeveloped country) and the imports of the developing country (exports of the developedcountry). On the basis of equations 8 and 12, we can write:

X = MiZ

0

pmi q¤

i L¤di =

JZ

i

pmi qiLdi

iL¤a¤J¤γJ¤ ¡ wL¤

iZ

0

aiγ idi = (J ¡ i)La¤JγJ ¡ L

JZ

i

a¤i γ idi

Re-writing the last term (of the right hand side) using equation 15, we obtain, aftersimpli…cation:

w =i£L¤a¤

J¤ γJ¤ + La¤JγJ

¤

L + (L + L¤)iR0

aiγidi

(17)

We have …nally obtained a system of four equations (14-17) with four unknowns, whichmust satisfy the following conditions: 0 < i < J < J¤.

Concerning the last expression, it should be noted that it is no easy task analyzing thistrade balance equation, as it includes many endogenous variables (w, i,J, J¤). Neverthe-less, we can make some comments on the partial e¤ects of the entry criterion terms (aiγ i),that which will highlight the potential impact of technical progress on trade equilibrium.Firstly, in the numerator, a¤

J¤ γJ¤ and a¤JγJ , refer to the threshold value of the marginal

consumed good in each country. According to equation 8, the spending on each good ispositively related to these terms. At the same time, according to equations 15-16, therange of consumed goods in each country decreases with these terms. Thus, a decreasein these boundary terms (through North’s technical progress) is likely to hurt South’srelative wage by lowering the spending devoted on its goods. On the contrary, a decreasein the integral value in the denominator re‡ects an higher gap between the thresholdvalue of the marginal consumed good (a¤

J¤ γJ¤ , a¤JγJ), and the threshold value of South’s

products (a iγ i). This is likely to result in higher spending on each of South’s products(equation 8), and then in an increase in the relative wage.

14 Concerning the structure of trade, we can also note that North is specialized in a wider range ofproducts. This characteristic comes directly from theorem 2 and 3 applied to the trade balance condition.It is in accordance with some recent empirical studies which highlight the fact that high income per capitacountries export a wider range of products. See, in particular, Hummels and Klenow’s paper (2005).

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At this point, it is again worthy to compare the mechanisms involved in the adjustmentof our trade balance equilibrium with those of Matsuyama and DFS. Looking at table 4,the di¤erences between the three models are straightforward15 . In DFS, the trade balanceequilibrium does not depend on technical coe¢cients. This is due to the Cobb Douglasfunction, according to which the share of spending on each good is …xed by the parameterβi. Every technical progress will have the e¤ect of increasing the quantity demanded inproportion to the fall in price. In Matsuyama, the trade balance equilibrium no longerdepends on the relative wage, but only on South’s technical coe¢cients. This comes fromthe use of the function of hierarchic desires, where the quantities demanded are constrainedto one. As North is specialized in lower priority goods, this entails that agents in the twocountries only use their increase in real income to demand goods produced by North.It follows that demand for South’s products is insensitive to price variation, and henceto wage variation. The only way to modify external demand for South is through itsown technical progress, enabling it to extend its range of specializations. In other words,under a function of hierarchic desires and an uniform distribution of income within eachcountry, di¤erences in income elasticities in trade are "apparent". They are determinedby the relative ability of each country to extend its range of traded goods16 . North isin this case more favored, as every change in real income is used to demand its (lowerpriority) goods. Contrary to our approach, non unitary income elasticities in trade, inMatsuyama’s model, are then not founded on the characteristics of the good itself, buton the assumption of an unconditional preference for North’s goods. This distinctionbetween the two models is at the heart of the di¤erences in the trade balance equations.

Finally, one should insist on the fact that the simultaneous inclusion of extensive andintensive consumption di¤erentiates strongly our model from these two approaches. Rel-ative wage is still a determinant of our trade balance equilibrium, because of demandsensitivity to price variation. Moreover, the inclusion of both countries’ technical coef-…cients in the trade balance equilibrium accounts for our endogenous elasticities. Thisre‡ects the impact of every technical progress on the share of spending devoted to eachcountry.

We now turn to the implication of our approach for the analysis of the internationaldi¤usion of welfare gains through trade. According to the above comparisons, we havedecided to devote a particular intention to the impact (on South’s welfare) of a North’stechnical progress biased toward the less priority goods in the case of a large technologicalgap. E¤ectively, we expect speci…c results in that con…guration at least for two reasons.Firstly, in our model, the size of the technological gap between the two countries will havea direct in‡uence on the value of trade elasticities, as the latter depends on the level ofincome of the agents. Secondly, the nature of the technical progress is also expected to becrucial because a good’s demand elasticity depends directly on its technical coe¢cient ofproduction. The analytical study of this case is conduced under speci…c functional formsand completed by simulations.

4 The e¤ect of a North’s technical progress on South’swelfare

4.1 The critical in‡uence of the size of the technological gap andthe nature of the technical progress

The choice of a functional form for a Ricardian inter-sectorial trade framework is not atrivial issue. In the standard interpretation of Dornbush, Fisher and Samuelson (1977), the

15 The main equations of the models of Matsuyama and DFS are presented in annex 3a.16 Following Krugman (1989), these aggregate elasticities of external demand can be viewed as "appar-

ent" because they do not refer to the evolution of demand for a given consumption basket, but to thechanges in the composition of traded goods.

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continuum of goods involves di¤erent sectors, the characteristics of which are not de…ned.The only precision made concerning sector characteristics is a ranking of goods alongthe continuum as a function of the relative productivity gap between the two countries.Krugman (1990) proposed an extension of the theoretical interpretation that can be madeof the DFS model: he argued that the ranking of goods according to the productivity gapcorresponds to the ranking of goods according to their technological content. However,choosing a functional form which can take into account the link between technologicalcontent and sector productivity is no easy task, insofar as that if the continuum involvesdi¤erent sectors, the respective units of each good are likely to be di¤erent. In otherwords, it is impossible directly to establish the relation between ai and i. In our model,we get around this problem by not directly …xing the functional form of ai , but only thatof aiγi , which we have de…ned as an increasing function of i (relation 13).In other words,we do not …x the primitives of the expressions for aiγ i and a¤

i γ i. We propose to use thefollowing functional forms:

aiγi = α + β.i (18)a¤

i γi = α¤ + β ¤.i (19)

In addition, our hypotheses about the technological gap will enable us to constrainparameter values. For an absolute productivity advantage increasing along the continuum,the su¢cient condition to allow South to be competitive in the …rst part of the continuumis:

ββ¤ >

αα¤ > 1

In this case, the ratio of production costs ( a¤i

ai) decreases along the whole continuum

with an asymptote tending to β¤

β .

We rewrite the general equilibrium model by applying the equations 18-19 to thesystem 14-17 and verify that an analytical solution exists17 . We then analyze the e¤ecton South’s utility of a biased technical progress (toward the less priority goods) in Northwhen the technological gap is large This con…guration is formally equivalent to a decreasein the value of β¤ when this parameter is very small (relatively to β). To simplify theanalysis, we study trade equilibrium around β¤ = 0.

To do this, we denote ε =p

β¤ and proceed to marginal developments of our equilib-rium equations around ε = 0. Results are the following ones :

i = ei(1 ¡ a.ε) (20)w = ew.(1 + b.ε) (21)

J =ejε.(1 + c.ε) (22)

J¤ =ej¤

ε.(1 ¡ d.ε) (23)

ei, ew,ej,ej¤, a, b, c and d are all positive parameters obtained after marginal developmentof equilibrium equations (their expressions are reported in annex 2c).

By including (20-23) in equation 2 we obtain an expression for South’s utility. Withthe same methodology (marginal developments), we can demonstrate that South’s utilitycan be represented, around ε = 0, by an equation of the following form :

U = A + B.ε (24)

where :17 See annexes 2a and 2b.

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A =αβ

. logα

(α + βei)+ei +

l(1 + l)

.1

(α + βei)

B = 1(α + βei).α¤ .(1 + l)

.

Ãej¤ ¡ ej1 + l

+ej3

!

In …gure 5, we have represented for speci…c value of parameters the equation 24 as wellas the general utility equation (2). This con…rms that equation (24) is good approximationof our utility function around ε = 0.

The two terms A and B are positive ( ej ¤ > ej). This implies that the utility of thedeveloping country is an increasing function of the term ε =

pβ¤. Hence, the result of

this analysis is that when the size of the technological gap increases (when ε tends toward0), the utility of the developing country decreases.

As a …rst comment, it is worthy to compare this result with those obtained by Mat-suyama (2000) and DFS (1977) under the same con…guration. In their model, the followingRicardian property is always veri…ed: as trade gains are derived from the existence of dif-ferences between countries, when these di¤erences grow larger, i.e. when technologicalshocks augment the existing terms of comparative advantage, the transaction gains in-crease1 8 . In other words, a biased technical progress in North improves South’s welfare.Actually, the di¤erence between these both approaches and our own model can be ex-plained on the basis of equation 20. E¤ectively, looking at equations 20-23, one can notethat around ε = 0, if the impact of β¤ on w and on J, J ¤ is standard (respectively positiveand negative), this is no more the case for i. The equation 20 implies that, the range ofSouth’s specialization is negatively related with the cost of production of North. Beforediscussing in more details the mechanisms at the heart of this "non standard" relation-ship, we aim to complete our analysis by looking for what happens outside the previouscon…guration. We proceed by simulations and relax …rst, the assumption of a very largetechnological gap, and second that of a biased technical progress.

For our simulations, we consider three di¤erent equilibriums, in order to measure theextent to which the qualitative results produced by the simulations are sensitive to the sizeof technological gap (simul 1, 2 and 3). These con…gurations di¤er in the value given to theparameter β¤. Table 5 presents the values taken by the di¤erent variables for these threeinitial equilibriums. We adopt a comparative static procedure and then study South’sutility in the case of a foreign technical progress biased towards the most sophisticatedgoods (FNUTS, variation of β¤), and in the case of a foreign uniform technical shock(FUTS, identical variation in α¤, β¤). These technological shocks are formally re‡ectedby a reduction in the parameters concerned by one half.

Simulations results con…rm that the size and the nature of the shock are two conditionsfor the appearance of a con…guration where South’s utility decreases as a result of atechnical progress in North. E¤ectively, by comparing tables 5 and 6, we can see thateven in the case of a very large technological gap, the South always bene…ts from North’stechnical progress when the latter is uniform along the continuum. Also, in the caseof a technical progress in the developed country growing over

£i, J¤¤ , the utility of the

developing country fall only under a large technological gap. This con…guration appearsfor simul 2 and 3 (compare tables 5 and 7). The existence of a threshold e¤ect after whichthe developing country may present a decrease of its utility appears clearly in …gure 6.

We now turn to the description of the main mechanisms embedded in our model whichare at the heart of the process of immeserizing in South after a technical progress in North.

18 Result of Matsuyama and DFS can directly be derived from their conditions of specialization (equation14) and their trade balance equilibrium (table 3). We have also reported graphically the impact of technicalshock in …gures of annex 3b.

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4.2 Mechanisms at the heart of an immeserizing e¤ect of externalgrowth

It is worthy to begin by an explanation of our results in the case of an uniform technicalprogress in the North. According to our simulations, in this con…guration, the classicale¤ect of welfare gain di¤usion holds, whatever the size of the technological gap. Thiscan be explained in the following way. Given uniform technical progress in the developedcountry, the productivity gap between the two countries increases uniformly. Accordingto the specialization equation (14), this entails a fall in the relative wage. But this fallstimulates demand for goods produced by the developing country, limiting the deterio-ration in its terms of trade. At the new equilibrium, the fall in the relative wage doesnot completely o¤set the initial productivity gains (…gures 7). Welfare increases in bothcountries. It can be noted that this result tallies with that of DFS, but di¤ers from thatof Matsuyama, where the developing country is totally impervious to the developed coun-try’s technical progress. The comparison between our result and that of Matsuyama isparticularly interesting. This highlights the fact that, despite unfavorable elasticities intrade, trade remains a good way to bene…t from external technical progress, once we haveallowed South to bene…t from price competitiveness.

Concerning now the di¤erences implied by a biased technical progress, the decreasein South’s utility can be explained in the following way. Our modeling of preferencesentails that the development gap (the di¤erence in per capita income) determines thedi¤erences in income and price elasticities between the two countries. We can make twoobservations about this e¤ect. Firstly, in a similar fashion to the uniform shock describedabove, the developing country is relatively penalized by its elasticities in trade. Secondly,according to theorem 6, we have seen that in our model, price elasticities increase alongthe continuum. Under a shock biased towards the lower priority goods, the goods whichbene…t from the highest price elasticities are then those which bene…t from the largestfalls in price. The price elasticity and income elasticity e¤ects contribute jointly to apronounced change in the distribution of expenditures in favor of the goods produced bythe developed country. The balance of trade equilibrium condition therefore requires afall in the relative wage, such that the developing country may see a fall in its aggregatereal income. As can be seen in …gures 8, the real costs of some consumed goods forthe developing country have risen. More precisely, the adverse impact of North’s biasedtechnical progress appears to be conduced by two e¤ects. Firstly, some goods which werepreviously produced by North are now produced by South, and consumed with higherreal prices. Secondly, the fall in the relative wage reduces South’s purchasing powerfor some of North’s goods. Actually, South does not bene…t from an reinforcement ofthe existing terms of comparative advantage, because the strong change in the spendingpatterns forces South to extend its specializations, despite the reduction in its comparativeadvantages (see tables 5 and 7; equation 22). The wider the development gap betweenthe two countries, the more this extension, necessary to return to the trade balanceequilibrium, will penalize the developing country in terms of changes in purchasing power.

One should also insist on another characteristic of our model : technical progresshurts the South despite the fact that South’s agents ultimately consume more sophisti-cated goods at lower prices (J increases). This result di¤ers strongly from Matsuyama’sapproach, where J can be viewed as a direct measure of utility. In our model, changes inagents’ utilities have to be apprehended through changes in the quantities and range ofgoods consumed. What happens here is that the increase in utility induced by a widerrange of consumed goods is o¤set by lower quantities of each good consumed. With thiskind of shock, there is actually a stronger change in the slope of the curve piγi

w , comparedwith the changes entailed by an uniform shock (see …gures 7 and 8). This change re‡ectsthe strong modi…cation in the consumption behavior of the agents, especially in their per-ception of the relative priority of each good. With a ‡atter curve, agents view goods thathad been previously considered as lower priority goods, as now having relatively morepriority. They extend the range of consumed goods accordingly. At the same time, the

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spending on relatively higher priority goods decreases (equation 8). This means that thefall in real income results in a reduction in the level of satisfaction provided by each good.In other words, in our model, changes in price structure indirectly account for changes inwhat we can call the "standard of consumption": South’s agents import a wider rangeof sophisticated goods (for instance, televisions or cell phones), to the detriment of thequantities of each good consumed (housing, clothes, etc.)19 .

Finally, it is worthy to brie‡y explain the reasons for which this con…guration doesn’tappear in the approaches of Matsuyama (2000) and DFS (1977). The di¤erence withour own model comes from the fact that, according to their assumption on preferences,in both other models the range of specialization does not increase as a result of thiskind of shock. Notably, as it has been already noted, in Matsuyama’s model the tradebalance condition is not modi…ed by biased foreign technical progress. Consequently, therange of South’s specializations remains constant. The fall in the relative wage is directlydetermined in equation 14 by the decrease in productivity of the marginal good i. Asthe productivity gains are increasing along

£i, J ¤¤ , this implies that South’s consumers

bene…t from an increase in their purchasing power for all goods after i. In the DFSmodel, North’s technical progress is accompanied by a reduction in South’s specializations.South’s consumers bene…t from an increase in welfare through the increasing purchasingpower gains along the goods continuum.

To conclude, the endogenous change in consumption and specialization in our modelcan be interpreted from the perspective of the product cycle theory20 . The biased technicalprogress in North has two main e¤ects on specializations. On the one hand, it stimulatesNorth’s production of sophisticated products, and on the other hand, it induces North togive up the production of less sophisticated goods (which are being produced by South).South’s access to more sophisticated goods increases, because of the standardization ofNorth’s production. At the same time, South becomes more competitive in some products,because of the increase in its relative poverty.

5 CONCLUSIONThis paper proposes a new framework for analyzing the changes in South’s welfare inducedby domestic and external technical progress. Our main contribution concerns the way wemodel consumption behavior. We have considered a hierarchic linear expenditure system.Goods can then be distinguished according to their order of entry into the consumptionbasket. This determines their relative degree of priority, and their respective incomeand price elasticity values. Moreover, we have established a link between these goods’characteristics and the technological dimension. In this way, agents’ perceptions of therelative priority of goods are dependent on the type of technical progress. We havecompared our results with those of Matsuyama (2000) and DFS (1977). As in our model,Matsuyama establishes a connection between technical progress, its income e¤ect, and thechange in spending patterns due to nonhomothetic preferences. This also highlights thedi¤erential impact of technical progress on welfare, depending on specialization patterns.

19 Some vertically di¤erentiated models also produce a con…guration where North’s technical progressmay be immiserizing for South. Nevertheless, the mechanisms involved are di¤erent. For instance,Stockey (1991) assumes a model where vertically di¤erentiated goods are imperfect substitutes. North’sagents then consume a wider range of quality goods. She shows that external progress is immiserizing forSouth, if it is biased toward non-traded goods (i.e. high quality goods consumed only by North). In ourmodel, this con…guration appears even if the technical progress bene…ts traded goods. Flam and Helpam(1987), consider a model with heterogenous population in each country. South (North) produces a low(high) quality good consumed by poor agents in the two countries. In this case, the global distributionof spending is dependant on the distribution of income. If the distribution of income shifts toward theagents who consume high quality goods, this entails a change of spending patterns in favor of North. Thesubsequent improvement in its terms of trade may be immiserizing for South. In our model, South canlose even under our assumption of a representative agent.

20 See for instance, the three stages of product developement in Vernon (1966, 1979), the models ofGrossman and Helpman (1991), and the empirical studies of Schoot (2004), Feenstra-Rose (2000), andHummels-Klenow (2005).

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The main contribution of our model, compared with that of Matsuyama, involves ourendogenous demand for quantities (and the subsequent endogenous income and priceelasticities). By giving up the constraint of the (0,1) consumption framework, we bringnew insights concerning the mechanisms at the heart of evolutions in welfare. Morespeci…cally, technical progress, by changing price structure, also modi…es the relativepriority of goods. This e¤ect is notably re‡ected by a change in the level of spending(devoted to each good), necessary to extend the range of consumption. We have calledthis e¤ect the endogenous change in the "standard of consumption". We have shownthat our model can present results that are a priori paradoxical in a Ricardian context:when there is technical progress in North biased towards the most technological goods,the greater the di¤erence between the two countries, the less the developing countrygains from trade. Technical progress in North may then be immeserizing for South. Thisimplies that there exists a level of development gap which maximize the welfare gainsfrom trade. This result can be viewed as the re‡ection of an adverse impact of a productcycle mechanism, which is driven by consumption behavior.

BIBLIOGRAPHYAMABLE B. (1992), E¤ets d’apprentissage, competitivité hors-prix et croissance cu-

mulative, Economie Appliquée, tome XLV, n±3.ATESOGLU H.S. (1994), An application of the kaldorian export-led model of growth

to the United States, Applied Economics, 26(5).ATESOGLU H.S. (1993), Exports, Capital Flows, relative prices and Economic Growth

in Canada, Journal of Post Keynesian Economics, 16.BAIRAM E.I and DEMPSTER G.J. (1991), The Harrod Foreign Trade Multiplier and

Economic Growth in Asian Countries, Applied Economics, 23(11).BAYOUMI T. (1999), "Estimating Trade Equations from Aggregate Bilateral Data",

IMF working paper, No. 99/74BERGSTRAND J.H. (1990), The Hecksher-Ohlin Samuelson Model, The Linder Hy-

pothesis and the Determinants of Bilateral Intra-Industry Trade, The Economic Journal,vol.100.

BHAGWATI J. (1958), Immiserizing growth: a geometrical note, Review of EconomicStudies, 25.

BILLS M. and KLENOW P.J. (2000),Quantifying Quality Growth, NBER Working-Paper Series, n±7695

CHUNG C. (2003), Non homothetic Preferences as a cause of Missing Trade and OtherMysteries, working paper Georgia Institute of Technology

DAVIS D.R. et WEINSTEIN D.E. [2001], An account of Global Factor Trade, Amer-ican Economic Review, 95(5)

DINOPOULOS E., FIJIWARA K. and SHIMOMURA K. (2006), International tradepatterns under quasi-linear preferences, working paper university of Florida

DORNBUSCH, S. FISHER and P.A. SAMUELSON (1977), Comparative Advantage,Trade, and Payments in a Ricardian Model with a continuum of Goods, American Eco-nomic Review, 67(5).

FAGERBERG J. (1988), International Competitiveness, Economic Journal, 98(391).FALKINGER J. and ZWEIMÜLLER J. (1996), The Cross-Country Engel Curve for

Product Diversi…cation, Structural Change and Economic Dynamics, vol.7.FEENSTRA R.C. (1994), "New Product Varieties and the Measurement of Interna-

tional Prices", The American Economic Review, vol.84.FEENSTRA R.C. et A. ROSE (2000), "Putting Things in Order : Trade Dynamics

and Product Cycles", Review of Economics and Statistics, MIT Press, vol.82(3).FILLAT C. and FRANCOIS J.F. (2004), "National and International Income Disper-

sion and Aggregate Expenditures", Tinbergen Institute discussion paper n.04-093/2.FLAM H. and HELPMAN E. (1987), Vertical product di¤erentiation and North-South

trade, American Economic Review, 77

14

hals

hs-0

0588

310,

ver

sion

1 -

22 A

pr 2

011

FOELLMI R.and ZWEIMULLER J. (2006) "Income Distribution and Demand-InducedInnovations," Review of Economic Studies, vol. 73(4)

FRANCOIS J.F. and KAPLAN S. (1996), Agregate Demand Shifts, Income distribu-tion and the Linder Hypothesis, The Review of Economics and Statistics, 78(2).

FUNKE M. and RUHNWEDEL R. (2001), "Export variety and export performance :empirical evidence from Easr Asia", Journal of Asian Economics, vol.12

GAGNON J.E. (2003), "Long-Run Supply and the Elasticities Approach to trade",International Finance Discussion papers, Board of Governors of the Federal Reserve Sys-tem

GROSSMAN G.M. and HELPMAN E. (1991), Innovation and Growth, MassachusettsInstitute of Technology Press, Cambridge.

HALLAK J.C. (2003), "Product Quality and the Direction of Trade”, Journal ofInternational Economics, (2006), 68(1)].

HOUTHAKKER H.S. and MAGEE S.P. (1969), Income and Price Elasticities in WorldTrade, The Review of Economics and Statistics, 51(2).

HUMMELS D.L. et P.J. KLENOW [2005], The Variety and Quality of Nation’s Trade,American Economic Review, 95

HUNTER L. (1991), "The contribution of nonhomothetic preferences to trade," Jour-nal of International Economics, vol. 30(3-4)

HUNTER L. and MARKUSEN J. (1988), Per Capita Income a Determinant of Trade,in R.Feenstra, ed. Empirical Methods and International Trade, MIT Press, Cambridge,Mass.

JACKSON L.F (1984), Hierarchic Demand and the Engel Curve for Variety, TheReview of Economics and Statistics, 66(1).

KALDOR N. (1970), The Case for Regional Policies, Scottish Journal of PoliticalEconomy, nov.

KRUGMAN P.R. (1990), A "Technology Gap" Model of International Trade, in Re-thinking International Trade, MIT Press, Cambridge.

KRUGMAN P.R. (1989), Di¤erences in income elasticities and trends in real exchangerates, European Economic Review, 33.

KUGLER M. and ZWEIMULLER J. (2002), International Trade when InequalityDetermines Agregate Demand, working paper 24, University of Southampton 23

MARKUSEN J.R. (1986), Explaining the Volume of Trade: An Ecletic Approach,American Economic Review, 76(5).

MARKUSEN J. and HUNTER L. (1988), Per capita Income as a basis for Trade,in Empirical Methods for International Trade, ed. R.C. Feenstra, Cambridge, MA MITPress.

MATSUYAMA K. (2000), A Ricardian Model with a Continuum of Goods under Non-homothetic Preferences: Demand Complementarities, Income Distribution, and North-South Trade, Journal of Political Economy, 108

MATSUYAMA K. (1992), Agricultural Productivity, Comparative Advantage andEconomic Growth, Journal of Economic Theory, 58.

MITRA D., TRINDADE V., (2005), Inequality and trade, Canadian Journal of Eco-nomics, Vol. 38, No. 4, pp. 1253-1271

Mc GREGOR P.G. and SWALES J.K. (1991), Thirwall’s Law and balance of paymentsconstrained growth: further comment on the debate, Applied Economics, 23.

Mc GREGOR P.G. and SWALES J.K. (1986), Balance of payments constrained growth:a rejoinder to Professor Thirlwall, Applied Economics, 18.

MELICIANI V. (2002), The Impact of technological specialisation on national perfor-mance in a balance-of-payments-constrained growth model, Structural Change and Eco-nomic Dynamics, 13(1).

MURPHY K.M., SHLEIFER A. and VISHNY R.(1989), Income Distribution, MarketSize, and Industrialization, The Quaterly Journal of Economics, August.

MUSCATELLI V., T.G. SRINIVASAN et D. VINES (1994), The empirical Modellingof NIE exports : an evaluation of di¤erent approaches, Journal of Development Studies,30.

15

hals

hs-0

0588

310,

ver

sion

1 -

22 A

pr 2

011

MUSCATELLI V., A. STEVENSON et C. MONTAGNA (1995), Modeling agregatemanufactured exports for some Asian newly industrialized economies, Review of Eco-nomics and Statistics, 77.

PASINETTI L. (1981), Structural Change and Economic Growth, Cambridge Univer-sity Press.

PUGA D., VENABLES A.J. (1999), Agglomeration and economic development : Im-port substitution vs. trade liberalisation, Economic Journal, 109

PERRATON J. (2003), Balance of Payments Constrained Growth and DevelopingCountries: an examination of Thirlwall’s hypothesis, International Review of AppliedEconomics, vol.17.

SCHOTT P.K. [2004], Across-Product versus Within-Product specialization in Inter-national Trade, Quaterly Journal of Economics, 119(2).

STIBORA J. and de VAAL A. (2006), Does Preferential Trade Bene…t Poor Countries?A General Equilibrium Assessment with Nonhomothetic Preferences, working paper

STIBORA J. and de VAAL A. ,(2007), Trade Policy in a Ricardian Model with aContinuum of Goods under Nonhomothetic Preferences, forthcoming. in Journal of De-velopment Economics,

STOKEY N.L. (1991), The Volume and the Composition of Trade Between Rich andPoor Countries, Review of Economic Studies, 58.

THIRLWALL A.P. (1979), The Balance of Payments Constraint as an Explanationof International Growth Rate Di¤erences, Banca Nazionale del Lavoro Quaterly Review,32(128).

TREFLER D.[1995], The Case of Missing Trade and Others Mysteries, AmericanEconomic Review, 85(5).

VERNON R. (1966), International Investment and International Trade in the ProductCycle, Quaterly Journal of Economics, 80.

YOUNG A. (1991), Learning by Doing and the Dynamic E¤ects of International Trade,The Quaterly Journal of Economics, may.

ZWEIMÜLLER J. (2000), Schumpeterian Entrepreneurs Meet Engel’s Law: The Im-pact of Inequality on Innovation-Driven Growth, Journal of Economic Growth, 5(2). 24

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TABLES

Table 1: Main characteristics of our demand functionIncome variation (y) Rank of good variation (i)

Range of products ∂J∂y > 0 T1

Share of spending ∂piqi∂y > 0 T2 ∂piqi

∂ i < 0 T3

Income elasticities ∂ηiy

∂y < 0 T4 ∂ηiy

∂i > 0 T5

Price elasticities∂jηi

pj∂ i > 0 T6

Cross-price elasticities ∂ηipk

∂ i > 0 T7

Table 2: Main properties of the demand function of the three modelsDFS (1977)C obb Do uglas

Mats (2000)Hierarchic des ires

Current modelHiera rchic purchases

ηiy = 1 0 ηi

y(y, i)ηi

p = 1 0 ηip(y, i)

J = J ¤ = cst J(y, p) J (y, p)qi = qi(y, p) 1 qi(y, p)

piqiy = βi = cst piqi

y (J, p) piqiy (J, qi , p)

Table 3: Structure of consumption and specializations in the two countriesSpecializations Price Country of production Country of consumption£

0, i¤

wai Developing country Both countries£i, J

¤a¤

i Developed country Both countries[J,J ¤] a¤

i Developed country Developed country

Table 4: Trade balance equilibrium conditionsDFS (1977) Matsuyama (2000)

wL1Ri

βi = L¤iR0

β i L = (L + L¤)iR

0aidi

Table 5: Initial equilibriums according the technological gapInitial equilibriums U U ¤ w i J J¤

(for each simul: α/α¤ = 2)Simul 1 (β/β ¤ = 6, 7) 6,7 15,0 0,31 2,6 14,2 25,7Simul 2 (β/β ¤ = 66, 7) 7,2 27,4 0,19 4,0 32,7 80,5Simul 3 (β/β ¤ = 1010, 1) 6.8 38.9 0.14 5.8 97.0 310.2

[α = 0.045; α¤ = 0.0225; β = 0.020; L = L¤ = 1]

Table 6: Equilibriums after foreign uniform technical shock (FUTS)Equilibriums after FUTS U U¤ w i J J ¤

(for each simul: α/α¤ = 4)Simul 1 (β/β ¤ = 13, 3) 6.9 23.6 0.16 2.1 14.6 36.4Simul 2 (β/β ¤ = 133, 3) 7.5 46.5 0.10 3.5 34.6 114.8Simul 3 (β/β ¤ = 2020, 2) 7.0 72,0 0,07 5,5 101,7 444,5

Table 7: Equilibriums after foreign non uniform technical shock (FNUTS)Equilibriums after FNUTS U U ¤ w i J J¤

(for each simul: α/α¤ = 2)Simul 1 (β/β ¤ = 13, 3) 7.0 18.6 0.26 2.9 18.0 36.2Simul 2 (β/β ¤ = 133, 3) 7.1 31.0 0.17 4.5 42.7 113.5Simul 3 (β/β ¤ = 2020, 2) 6,77 40,6 0,13 6,1 130,6 438,0

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FIGURES

Figure 1: Range of consumed goods according to the entry criterion ( 1piγi

)

1/(piγi)

λ

J

(0, J) represent the segment of consumed goods under the assumption of a decreasingmonotonic function for 1

piγi

Figure 2: Proof of theorem 1:The number of consumed goods J is an increasingfunction of y

J

piγi pJγJ

piγi

y

∫i∈K piγidi

As y increases, the area of the triangle above the curve piγ i increases, which is expressedby an increase of consumed goods.

Figure 3: The change of the quantities and varieties consumed with income variation

piγi

piγi

J J’ i

pJγJ

p’JγJ

piγi

piγi

J i

pJγJ

Along the continuum, the distance between pJγJ and piγi indicates the spendingdevoted to each good.

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Figure 4: The impact of prices change on the composition of consumption

p iγ i

p ’ iγ i

p iγ i

J J ’ i

p Jγ J p ’ Jγ J

Figures 5: South’s utility

The general function (equation 2) is represented by the normal line and the border case(around ε = 0) by the doted line (equation 24).

Figure 6: Utility in developing country and technological gap

6,4

6,6

6,8

7,0

7,2

7,4

7 19 55 157 451

The vertical axis represents the utility of developing country and the horizontal axisreports the value taken by β/β¤ (under the asumptions α/α¤ = 2 and constant)

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Figures 7: E¤ect of FUTS on real prices of goods (pmiw ) consumed by devel-

oping country21

FUTS / Simul 1

FUTS / Simul 2

FUTS / Simul 3

21 The horizontal axis corresponds to the continuum of goods (i). The normal line represents the realprice of consumed goods in South before the shock. The doted line represents the real price of these goodsafter the shock. The break in the lines refers to i.

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Figures 8: E¤ect of FNUTS on real prices of goods ( pmiw ) consumed by

developing country

FNUTS / Simul 1

FNUTS / Simul 2

FNUTS / Simul 3

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ANNEXES

ANNEX 1: Demonstrations of seven theorems on consumption

Theorem 1 : The number of goods consumed J is an increasing function of yThis can be demonstrated graphically using equation 7 (…gure 2). When y increases,

the number of goods consumed J increases, as does its threshold value pJγJ . This isexpressed by the increase in the area of the triangle above the curve piγi . For given pricesof goods, the values of variables pJ γJ and J therefore represent a wealth e¤ect.

Theorem 2 : The amount spent on each good piqi is an increasing function of yThis is proved simply by applying theorem 1 in equation 8: as J and therefore pJ γJ

are an increasing function of y, it follows that the amount spent on each good piqi is alsoan increasing function of y.

Theorem 3 : The amount spent on each good piqi is a decreasing function of iThis can also be demonstrated on the basis of equation 8, according to which, the

amount spent on a good piqi is given by the gap between the threshold e¤ect value of themarginal good and the threshold e¤ect value of the good i. The amount spent on eachgood is therefore a decreasing function of that good’s position in the continuum.

Theorem 4: Income elasticity ηiy is a decreasing function of y

This is proved by using equation 9 to calculate the expression pkγk of good k whichsatis…es the condition ηk

y = 1. To prove this theorem, it is then su¢cient to verify thatwhen income increases, this good corresponds to a higher index value in the continuum,∂pkγk/∂y > 0. Given that income elasticities increase along the continuum (theorem 5),we denote k (ηk

y = 1) the good which marks the limit between the "luxury goods" segment(ηi

y > 1) and the "necessity goods" segment (η jy < 1). Using equations 8 and 9, and after

transformation (using equation 7), we can write for this good:

pkγk =

JR0

piγ idi

J(25)

To prove theorem 4, we must verify that when income increases, the good k (whichsatis…es the condition ηk

y = 1) corresponds to a higher index value in the continuum ofgoods( ∂(pkγk)

∂ y > 0). Using partial derivative, one can …rst deduce from theorem 1 that∂J∂y > 0. Second, it is straightforward to show that when we calculate the partial derivativeaccording to J and simplify the result (using equation 7), we obtain:

∂(pkγk )∂J

=yJ 2 > 0

Theorem 5 : Income elasticity ηiy is an increasing function of i

This theorem is directly proved by applying theorem 3 to equation 9.

Theorem 6 : The absolute value of price elasticity¯̄ηi

p¯̄is an increasing function of i

This can be directly derived from the application of theorem 3 to equation 10.

Theorem 7 : The substitution e¤ect is an increasing function of i (for a given changein the price of a good k)

This is proved directly by applying theorem 3 to equation 11.

ANNEX 2a: New equilibrium conditionsOn the basis of equations (14-17) and (18-19), we can write :

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w =α¤ + β¤.iα + β.i

(26)

J 2 = i2 +wβ¤ (2 ¡ βi2) (27)

J ¤2 = i2 +(2 ¡ wβi2)

β ¤ (28)

Consequently, it is indeed veri…ed that J ¤ > J when w < 1.On the basis of these three equations, we can determine a polynomial for i:

P (i) =

α¤Ã

l ¡ (1 + l)βi2

2

!

+β¤iµ

l + (1 + l)i(α +βi2

) ¡¡α + βi

¢(J ¤ + lJ )

¶(29)

= 0

With l = LL¤

ANNEX 2b: Veri…cation of the existence of an analytical solutionWe shall now demonstrate that the polynomial which determines the expression of i

presents at least one economically possible solution (i > 0). To do so, we simply need toverify that the solution can be ‡anked by two values (one positive and one negative). Wetherefore ‡ank P (i) by two extreme values of i:

- The developing country does not specialize in any good (i = 0). If we de…nei = i0 = 0, then equation 29 can be written:

P (i0) = α¤l > 0 (30)

- The developed country satis…es all its needs through its domestic production(i = J). According to the expression of the marginal good consumed by the domestic

economy (equation 27), the value of i corresponding to this con…guration is i = i1 =q

2β .

With this value, we obtain:

P (i1) = ¡α¤ + β¤i1l + β¤ 2β

(1 + l)(α +βi12

) ¡ β¤i1¡α + βi1

¢(J¤ + lJ ) (31)

To determine the sign of this polynomial, we proceed as follows:With i1 =

q2β , we know that i = J and J ¤ > J = i1.

Consequently, we can write the following relation for the last term of the polynomial:

¡β¤i1¡α + βi1

¢(J ¤ + lJ ) < ¡β¤i1

¡α + βi1

¢(i1 + i1l)

¡β¤i1¡α + βi1

¢(J ¤ + lJ ) < ¡β¤ ¡

i1¢2 ¡

α + βi1¢(1 + l)

¡β¤i1¡α + βi1

¢(J ¤ + lJ ) < ¡β¤ 2

β(1 + l)α ¡ 2β¤i1(1 + l)

By substituting the right-hand expression for the last term of equation 31, we verify:

P (i1) < ¡α¤ + β¤i1(1 + 2l) + β¤ 2β

(1 + l)α ¡ β¤ 2β

(1 + l)α ¡ 2β¤i1(1 + l)

P (i1) < ¡α¤ ¡ β¤i1 < 0

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So, by ‡anking i in this way, we have demonstrated that our polynomial has at leastone root and that this root, with a value between i0 and i1, is economically possible.

ANNEX 2c : Equilibrium around ε = 0

By proceeding to marginal development of equations 26-29, we have obtained equations20-23. Their respective parameters can be written in the following way:

ei =µ

2.lβ(1 + l)

¶1/2

(32)

ew =α¤

α + β.ei(33)

ej =µ

2. ew1 + l

¶1/2

(34)

ej¤ =µ

2 ¡ 2ew.l

1 + l

¶1/2

(35)

and :

a =ej¤ + l.ejew(1 + l)β

.1ei

(36)

b =a.β. ew

α¤ .ei (37)

c =a.β2

.ei.µei.(1 + l) +

ewα¤

¶(38)

d =a.ei2.β. ew

ej ¤2 .

Ã1 +

eiβ. ew2.α¤

!(39)

ANNEX 3a: The equations of the DFS and Matsuyama modelsThe supply side hypotheses are similar to our own model. Consequently, the condition

of specialization, i.e. equation 14, is identical in all three models. The di¤erences in thechoice of utility function, on the other hand, induce di¤erent equations for the balanceof trade equilibrium. To make it easier to compare the three models, the equilibriumequations are presented under the hypothesis βi = 1.

DFS (1977) Matsuyama (2000)

Max U =JR0

βi ln qidi Max U =

1Z

0

βiqidi (A1)

sc w =JR0

piqidi sc w =

1Z

0

piqidi (A2)

Xi = pmi q¤

i .L¤ = 1J .L¤ Xi = pm

i q¤i L¤ = waiL¤ (A3)

Mi = pmi qi.L = w

J .L Mi = pmi qiL = a¤

i L (A4)

J = J¤ = cste wL = LwiR

0aidi + L

JRi

a¤i di (A5)

L¤ = L¤wiR0

aidi + L¤J¤Ri

a¤i di (A6)

X=M )

wLiR0

βi = L¤1Ri

β i

X=M )

L = (L + L¤)iR0

aidi (A7)

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- (A1) and (A2) represent the agent’s maximization programme.- (A2) and (A3) represent the demand for imports per product addressed to the devel-

oping country and the demand for exports that this country addresses to the developedcountry.

- The balance of trade equilibrium condition (A7) is obtained through (A2) and (A3).For Matsuyama, this expression is simpli…ed with the help of equation (A5).

- (A5) and (A6) represent the exhaustion of the budget constraint in the two countriesand makes it possible to determine the number of varieties consumed in Matsuyama’smodel.

ANNEX 3b: Comparison of the e¤ect of technological shocksThe line A represents the condition of specialization (equation 14) and the line B

represents the balance of trade equilibrium condition (equations A7). B’ and A’ representthe shifting of these lines with the occurrence of technical progress (TP). It should benoted that the graphic representation proposed here is deliberately simpli…ed2 2 . It shouldbe read in the following manner:

- Along the y axis, the shift from one equilibrium to another gives the size of thevariation in relative wage

- The distance between A and A0, at any given point on the x axis, tells us the size ofthe productivity gains for a good i.

From the comparison of these two distances, we can deduce the variation in the pur-chasing power of an agent for every good consumed.

Dornbush, Fisher, Samuleson (1977)

Matsuyama (2000)

Uniform technical progress in North (FUTS)

Biased technical progress In North (FUTS)

B

A

w

A’

i

w

B

A’

A

i

i

B

A

w

A’

w B

A’

A

i

22 This presentation is intuitive. For more rigour, the reasoning in terms of variations would requiremodi…cation of the scale in logarithm or the representation of shifts in a non-linear form. This wouldcomplicate the presentation without modifying the results.

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